Optimal. Leaf size=131 \[ -\frac {b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}+\frac {b \sqrt {x} (7 b B-5 A c)}{c^4}-\frac {x^{3/2} (7 b B-5 A c)}{3 c^3}+\frac {x^{5/2} (7 b B-5 A c)}{5 b c^2}-\frac {x^{7/2} (b B-A c)}{b c (b+c x)} \]
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Rubi [A] time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 50, 63, 205} \begin {gather*} -\frac {b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}+\frac {x^{5/2} (7 b B-5 A c)}{5 b c^2}-\frac {x^{3/2} (7 b B-5 A c)}{3 c^3}+\frac {b \sqrt {x} (7 b B-5 A c)}{c^4}-\frac {x^{7/2} (b B-A c)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac {x^{5/2} (A+B x)}{(b+c x)^2} \, dx\\ &=-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}-\frac {\left (-\frac {7 b B}{2}+\frac {5 A c}{2}\right ) \int \frac {x^{5/2}}{b+c x} \, dx}{b c}\\ &=\frac {(7 b B-5 A c) x^{5/2}}{5 b c^2}-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}-\frac {(7 b B-5 A c) \int \frac {x^{3/2}}{b+c x} \, dx}{2 c^2}\\ &=-\frac {(7 b B-5 A c) x^{3/2}}{3 c^3}+\frac {(7 b B-5 A c) x^{5/2}}{5 b c^2}-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}+\frac {(b (7 b B-5 A c)) \int \frac {\sqrt {x}}{b+c x} \, dx}{2 c^3}\\ &=\frac {b (7 b B-5 A c) \sqrt {x}}{c^4}-\frac {(7 b B-5 A c) x^{3/2}}{3 c^3}+\frac {(7 b B-5 A c) x^{5/2}}{5 b c^2}-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}-\frac {\left (b^2 (7 b B-5 A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 c^4}\\ &=\frac {b (7 b B-5 A c) \sqrt {x}}{c^4}-\frac {(7 b B-5 A c) x^{3/2}}{3 c^3}+\frac {(7 b B-5 A c) x^{5/2}}{5 b c^2}-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}-\frac {\left (b^2 (7 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^4}\\ &=\frac {b (7 b B-5 A c) \sqrt {x}}{c^4}-\frac {(7 b B-5 A c) x^{3/2}}{3 c^3}+\frac {(7 b B-5 A c) x^{5/2}}{5 b c^2}-\frac {(b B-A c) x^{7/2}}{b c (b+c x)}-\frac {b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 110, normalized size = 0.84 \begin {gather*} \frac {\sqrt {x} \left (b^2 (70 B c x-75 A c)-2 b c^2 x (25 A+7 B x)+2 c^3 x^2 (5 A+3 B x)+105 b^3 B\right )}{15 c^4 (b+c x)}-\frac {b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 119, normalized size = 0.91 \begin {gather*} \frac {\left (5 A b^{3/2} c-7 b^{5/2} B\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}+\frac {\sqrt {x} \left (-75 A b^2 c-50 A b c^2 x+10 A c^3 x^2+105 b^3 B+70 b^2 B c x-14 b B c^2 x^2+6 B c^3 x^3\right )}{15 c^4 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 290, normalized size = 2.21 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{3} - 5 \, A b^{2} c + {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x + 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \, {\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {x}}{30 \, {\left (c^{5} x + b c^{4}\right )}}, -\frac {15 \, {\left (7 \, B b^{3} - 5 \, A b^{2} c + {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) - {\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \, {\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {x}}{15 \, {\left (c^{5} x + b c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 122, normalized size = 0.93 \begin {gather*} -\frac {{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {B b^{3} \sqrt {x} - A b^{2} c \sqrt {x}}{{\left (c x + b\right )} c^{4}} + \frac {2 \, {\left (3 \, B c^{8} x^{\frac {5}{2}} - 10 \, B b c^{7} x^{\frac {3}{2}} + 5 \, A c^{8} x^{\frac {3}{2}} + 45 \, B b^{2} c^{6} \sqrt {x} - 30 \, A b c^{7} \sqrt {x}\right )}}{15 \, c^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 139, normalized size = 1.06 \begin {gather*} \frac {2 B \,x^{\frac {5}{2}}}{5 c^{2}}+\frac {5 A \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}-\frac {7 B \,b^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{4}}-\frac {A \,b^{2} \sqrt {x}}{\left (c x +b \right ) c^{3}}+\frac {2 A \,x^{\frac {3}{2}}}{3 c^{2}}+\frac {B \,b^{3} \sqrt {x}}{\left (c x +b \right ) c^{4}}-\frac {4 B b \,x^{\frac {3}{2}}}{3 c^{3}}-\frac {4 A b \sqrt {x}}{c^{3}}+\frac {6 B \,b^{2} \sqrt {x}}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 115, normalized size = 0.88 \begin {gather*} \frac {{\left (B b^{3} - A b^{2} c\right )} \sqrt {x}}{c^{5} x + b c^{4}} - \frac {{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {2 \, {\left (3 \, B c^{2} x^{\frac {5}{2}} - 5 \, {\left (2 \, B b c - A c^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (3 \, B b^{2} - 2 \, A b c\right )} \sqrt {x}\right )}}{15 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 146, normalized size = 1.11 \begin {gather*} x^{3/2}\,\left (\frac {2\,A}{3\,c^2}-\frac {4\,B\,b}{3\,c^3}\right )-\sqrt {x}\,\left (\frac {2\,b\,\left (\frac {2\,A}{c^2}-\frac {4\,B\,b}{c^3}\right )}{c}+\frac {2\,B\,b^2}{c^4}\right )+\frac {2\,B\,x^{5/2}}{5\,c^2}+\frac {\sqrt {x}\,\left (B\,b^3-A\,b^2\,c\right )}{x\,c^5+b\,c^4}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,\sqrt {x}\,\left (5\,A\,c-7\,B\,b\right )}{7\,B\,b^3-5\,A\,b^2\,c}\right )\,\left (5\,A\,c-7\,B\,b\right )}{c^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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